CHAOS MODELS IN ECONOMICS
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1 Vlad Sorin CHOS MODELS IN ECONOMICS fan cl Mar Univrsiy of Sucava, Economic Scincs and Public dminisraion Faculy, Univrsi ii no., Romania, 79, / bsrac Th papr discusss h main idas of h chaos hory and prsns mainly h imporanc of h nonlinariis in h mahmaical modls. Chaos and ordr ar apparnly wo opposi rms. Th fac ha in chaos can b found a crain prcis symmry (Fignbaum numbrs) is vn mor surprising. s an illusraion of h ubiquiy of chaos, wo modls among many ohr xising modls ha hav chaoic faurs ar prsnd hr: h nonlinar fdback profi modl and on modl for h simulaion of h xchang ra. Kywords: chaos, nonlinar sysms, complx bhavior, bifurcaion diagram Inroducion On of h axioms of h modrn scinc assrs ha if an accura dscripion of a physical sysm can b idnifid hn h possibiliy of a dpr undrsanding of h sysm and h prdicion of h sysm voluion is possibl. Ths assrions ar no always corrc. For insanc, if on applis h laws of moion sad by Nwon, hn hr is possibl o prdic xacly h orbi of h Moon around h Earh if h influnc of ohr plans is no considrd. Ths prdicions wr vrifid and provd o b accura. If h hird plan is includd, h mahmaical modl of h inracion of h wo bodis bcoms h hr bodis problm, solvd by Nwon bu for a limid s of cass and unsolvd for h gnral cas. Today by mans of a compur, h r bodis problm can b solvd, bu on can obsrv ha h prdicion of h orbi of h hird plan is ofn impossibl. larg numbr of ral sysms hav a nonlinar bhavior dspi h idalizd linar bhavior usd in modling. Th dvlopmn of a nw way of daling wih nonlinar sysms is obvious. This nw way of daling xiss alrady dspi h fac ha h sudy of h nonlinariy is sill a h bginning. Som changs in nonlinar sysms can lad o a complx and rraic bhavior calld chaos. Th nonlinariy is on of h condiions ndd by a sysm in ordr o dvlop chaos. Th rm chaos is usd o dscrib h bhavior of a sysm ha is apriodic and apparnly random. S. H. Srogaz dfins chaos as an apriodic long im bhavior dvlopd by a drminisic sysm highly snsiiv on iniial condiion. [] Bhind his apparnly random bhavior lis h drminisic characr drmind by h quaions dscribing h sysm. Mos of h sysms ha ar usd as xampls o xplain h concps of chaos hory ar drminisic. Thr ar wo yps of chaos: drminisic and nondrminisic. Th drminisic chaos rprsns h chaoic moion of h nonlinar sysms whos dynamic laws drmins uniquly h voluion of h sysm s sa basd on h prvious voluion. Th drminisic chaos rprsns only on paricular cas of wha is calld nondrminisic chaos ha xhibis a suprxponnial divrgnc of h rajcoris. In his cas h quaions dscribing h voluion of h sysm ar no known. Th boh ways of chaos manifsaions ar shor-rm prdicabl bu long rm unprdicabl. Th chaos and h concps rlad o h dynamics of h sysms and h hir modling using diffrnial quaions is namd h chaos hory and is ighly rlad wih h noion of nonlinariy [4]. Th nonlinariy implis h loss of h causaliy corrlaion bwn h prurbaion and ffc propagad in im. Th sudy of h nonlinariy is namd nonlinar dynamics a capivaing domain using a mahmaical apparaus sill undr dvlopmn. Dspi h fac ha h idas lading o h mrgnc of h chaos hory xisd bfor longim, Lornz (96) crad a mahmaical modl of h convcion currns circulaion in amosphr and obsrvd ha whn h sysms bgins wih iniial condiions slighly changd from h prvious ons, h rsuls ar complly diffrn. This phnomnon will li a h basis of a vry popular paradigm of chaos namd 955
2 h burfly ffc, ha sas ha if h flapping of a burfly slighly modifis h amosphrically condiions in h mazonian jungl, his fac can hav an impac, a h nd of a complx caus ffc chain in sing off a ornado in Txas. Th burfly ffc paradigm conains h ssnc of h phnomnon characrizing h chaos: firs, h snsiiv dpndnc on iniial condiions and scond h fac ha o prdic h fuur sa of a chaoic sysm, h currn sa nd o b known wih infini prdicion. Th manifsaion of chaos can b found vrywhr in h ral world, for insanc: h propagaion of h avalanchs, pidmics sprading, clima voluion, har bas, lasrs, lcronic circuis, c. Figur. Lornz aracor h burfly of chaos hory. lgiima qusion is ha h chaos is h rul or h xcpion from h rul. Taking ino accoun ha mos of h sysms of h ral world ar nonlinar (h basic condiion for h mrgnc of chaos), sms ha chaos could b on of h no so obvious faurs of h naur. Th imporanc of sudying chaos is ha chaos offrs an alrna mhod ha xplains h apparnly random bhavior of h complx sysms. Th chaos plus h spcific mahmaical ools is a framwork of sudying diffrn modls from diffrn filds, modls ha can b rducd o lmnary modls wih known chaoic bhavior for som valus of h paramrs. Th way o chaos bgins wih h phnomnon of priod doubling. Th priod doubling volvs in, 4, 8, 6 and so on priods and h sysm voluion can abruply fall ino chaoic rgim. In h cas of unimodal funcion hr is an inrsing symmry in h paramr valus for wha h priod doubling occurs. If is h valu of h conrol paramr for wha h firs priod doubling occurs and n is h valu for wha h n h priod doubling occurs, hn: δ = lim n n n+ n n = whr δ is h Fignbaum numbr valabl for all unimodal funcions.[5] () 956
3 Nonlinar Modls. Chaos in xchang ras For h simulaion of h volail bhavior of h xchang ras wr crad modls ha ra h xchang ras as bing prics of h financial assssmns radd on fficin marks. Th currn xchang ra conains h currnly availabl informaion and h changs obsrvd rflc h ffc of h nw vns ha ar unprdicabl by dfiniion. Th hory sas ha an accura a priori prdicion of h xchang ra voluion is impossibl o b mad bu h subsqun xplanaion of h changs is possibl. In ordr o limina hs difficulis, h chaos hory and h nonlinar modls ar xnsivly usd. Th firs rsarchs hav bn carrid ou saring from 98. In h majoriy of siuaions hs modls ar highly nonlinar and rsul in a wid rang of dynamic bhavior, including chaoic dynamics. Thr is a dispu ovr h manifsaion of chaoic dynamics in xchang ras. Thr ar many sudis ha ar posiiv o h chaoic dynamics (Fdrici, Wsrhoff, Darvas 998, Homms 5, Vandrocicz 6) and also a numbr of sudis ha ar rjcing h chaos in xchang ra (Brooks, Srlis). Th chaos hory dmonsras ha vn h simpls dynamical sysms can xhibi a som poin a vry complx bhavior. If h xchang ras variaion is causd du o h chaoic naur of h sysm, his should lad o h fac ha h smalls influncs should hav h ffc of a nonlinariy ovr h xchang ras xacly wha happns in raliy. Th firs modl prsnd dmonsras h fac ha vn h simpls modls can xhibi chaoic bhavior. [] Th dmand of forign currncy is drmind as prcnag of h dviaion of currn xchang ra owards h xpcd on.[] whr is h domsic pric of h forign currncy is h fuur simad xchang ra S = α, α () α is h snsiiviy paramr Th rad balanc (T i ) is a linar funcion dpnding on h currn xchang ras and h corrsponding xchang ra for h las priod, wrin as dviaion from h xpcd valus and is givn by h quaion: Th xpcd xchang ra rprsns h sabl sa a which h spculaors on h mark do no wish o sll nor buy. ( ) + γ( ) β, γ > T = β () Th claring of h xchang marks wris as: fr rplacing quaions () and () in (4), quaion (4) bcoms: S = T (4) [( β + γ) γ a ] α β = (5) Th quaion 5 has wo roos, h posiiv on bing considrd for obvious rasons. Th rsuling nonlinar quaion is: 957
4 = [( β + γ) γ a ] [( β + γ) γ a ] β β β α for α=β=4 and γ=6. Th graphical rprsnaion of h soluion show ha h graph prsns a pak valu of.76 and a minimum valu of.9. ny ohr valu from ousid h inrval rprsnd by hs wo valus is aracd. Th voluion of h sysm wih h spcifid paramrs is chaoic bcaus saisfis h Ly- York condiion []. Th Figur illusras h voluion of h sysm for wo iniial slighly diffrn valus:. and.5 (h dod lin). Th valus of h wo im sris ar idnical for a shor priod of im (h firs iraions) and hn h rajcoris of h sysms ar divrging. (6) Figur. Th influnc of h iniial condiions. Th scarplos for h wo im sris ar providd o dmonsra h indpndnc of h wo im sris afr iraions. Th scarplos prsnd in Figur and Figur 4 on of h fingrprins of chaos: h disanc bwn wo rajcoris saring from narby poins in h sa spac divrg ovr im. Cycls (Iraions) - Cycls (Iraions) -5 x()= x()=. x()= x()=. a 958 b
5 Figur a. Th scarplo for h firs iraions and b) h scarplo for h las 4 iraions. Whn h snsiiviy paramr is varid, h sam ffcs can b obsrvd. Figur 4 prsns h rajcoris of h sysm for wo vry nar valus of α. valu.5.5 alfa=4 alfa= im Figur 4. Th influnc of changing snsiiviy paramr. Th apparnly irrlvan changs can affc h longim bhavior of h xchang ra modld using h Ellis modl and som of hs small shocks can drmin h sysm o fall ino h chaoic rgim. B. Th modl of h nonlinar fdback mchanism of h profi Th currn spnding of a firm can influnc h valu of h profi obaind a h nd of h rfrnc priod. Th profi will influnc h spnding ovr h nx priod. Th dpndnc bwn h prvious valu of h profi and h currn valu is nonlinar bcaus an incras of h spnding dos no rflc in an incras of h profi. Th law of h dcras of h fficaciousnss assrs ha a crain man valu rachs minimum or imum valu whn is magniud quals h marginal valu. On can invs in a crain producion capabiliy bu his dosn guaran an unlimid incras of h producion bu h incras up o a crain poin. Byond ha poin h incras of h invsmn dos no gnras a corrsponding incras of h producion. Th dpndnc bwn h currn profi and h prvious profi can b modld by using h quaion: Th imum profi Dividing h quaion (6) wih + = B (6).is supposd ha i can b drmind. L π = and h quaion (7) bcoms: B h following rsul is obaind: = B (7) B π + = π π (8) If w ak = h quaion abov bcoms h logisic quaion: = π π = ( π ) π π + (9)
6 Th logisic map xhibis h sam dpndnc on h iniial condiion: h slighs chang of h iniial condiion causs a complly diffrn voluion. Th complx bhavior of h apparnly simpl funcions can b obsrvd using h bifurcaion diagram. Th bifurcaion diagram (Figur 5) is an xclln ool allowing analyzing h bhavior of a funcion by varying a conrol paramr (in h cas of logisic funcion, h conrol paramr is ). Th logisic funcion is known o hav a chaoic bhavior wih small isls of priodiciy for a valu of h paramr grar ha.57. For [.57, 4] hr ar small aras of priodiciy, h whi srips ha can b obsrvd in h figur. For >4 h bhavior is complly chaoic. Figur 5. Th bifurcaion diagram for h logisic funcion Conclusions Chaos is can b found almos vrywhr in h naur. Chaos hory and fracals ar currnly applid in h sudy of h naural phnomnon. n ssnial condiion ndd in ordr ha chaos o mrg is o hav nonlinar sysms. In fac vry fw of all modls ar purly linar, h vas majoriy of h sysms ar nonlinar. Th papr mphasizs wo of h faurs of h chaoic sysms: dpndnc o iniial condiions and h divrgnc of narby rajcoris. Two of h modls usd in conomy ha could xhibi chaos ar dscribd and discussd. Rfrncs. E. Prs Chaos and Ordr in h Capial Marks. Nw York: John Wily & Sons, J. Ellis, Non-linariis and chaos in xchang ras, in Chaos and Non-Linar Modls in Economics: Thory and pplicaions, pp.87-95, Edward Elgar Publishing, R.C. Hilborn, Chaos and Nonlinar Dynamics, Oxford Univrsiy Prss, S. Kucha, Nonlinariy and Chaos in Macroconomics and Financial Marks 5. S. H. Srogaz, Nonlinar dynamics and chaos wih applicaions o Physics, Chmisry and Enginring, Prsus Books,
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